Now we will apply the Hilbert expansion (c.f. [1]}) to the scaled BFD equation (REF ). In the process, the arguments are largely formal but rational, hence highly instructive. To this end, the bilinear operator
\(\mathcal {Q}(F,G)=\iint _{\mathbb {R}^3\times \mathbb {S}^2}|(v_*-v)\cdot \omega |\left(F_*^\prime G^\prime -F_* G\right)\mathrm {d}\omega \mathrm {d}v_*,\) 
